A Bulk Queuing Model of Optional Second Phase Service with Short and Long Vacations (original) (raw)
Related papers
Working vacations queueing model with multiple types of server breakdowns
Applied Mathematical Modelling, 2010
This paper deals with a single server working vacation queueing model with multiple types of server breakdowns. In a working vacations queueing model, the server works at a different rate instead of being completely idle during the vacation period; the arrival rate varies according to the server's status. It is assumed that the server is subject to interruption due to multiple types of breakdowns and is sent immediately for repair. Each type of breakdown requires a finite random number of stages of repair. The life time of the server and the repair time of each phase are assumed to be exponentially distributed. We propose a matrix-geometric approach for computing the stationary queue length distribution. Various performance indices namely the expected length of busy period, the expected length of working vacation period, the mean waiting time and average delay, etc. are established. In order to validate the analytical approach, by taking illustration, we compute numerical results. The sensitivity analysis is also performed to explore the effect of different parameters.
Finite Capacity Queueing System with Vacations and Server Breakdowns
International Journal of Engineering, 2011
This paper deals with finite capacity single server queuing system with vacations. Vacation starts at rate if the system is empty. Also, the server takes another vacation if upon his arrival to the system, finds the system empty. Customers arrive in the system in Poisson fashion at rate 0 during vacation, faster rate f during active service and slower rate 0 s during the breakdown. Customers are served exponentially with the rate . Server breakdowns at rate b and it is immediately repaired exponentially with the rate r. We derive the explicit formulas for queue length distribution, average queue length, average number of customers in the system and average waiting time for a customer in queue, and in the system. Numerical illustrations have been cited to show that the proposed model is practically sound.
A Batch Arrival Queue with Second Optional Service and Reneging During Vacation Periods
2013
We study a two phase queuing system model where arrivals come to the system in batches of variable size following a compound Poisson process. We consider that service is provided in two phases, the first service is essential and second service is optional. Service becomes unavailable when the server goes for vacation and customers may decide to renege. We treat reneging in this paper when service is unavailable as the server is on vacation. We obtain steady state results in terms of probability generating function. Some special cases are discussed and a numerical illustration is provided. KEY WORDS: Second optional service, Reneging, Server vacation, Steady state queue size distribution MSC: 90B22 RESUMEN Estudiamos un sistema bifásico de colas donde los arribos llegan al sistema en lotes de tamaño variable que siguen un proceso compuesto de Poisson. Consideramos que el servicio esta provisto de dos fases, en el primero el servicio es esencial y el segundo el servicio es opcional. E...
2013
This paper investigates a single server queue with Poisson arrivals, general (arbitrary) service time distributions and Bernoulli vacation subject to random breakdowns. However, after the completion of a service, the server will take Bernoulli vacation, that the server make take a vacation with probability θ or may continue to stay in the system with probability 1 − θ for serving the next customer, if any. In addition to this, the vacation period of the server has two phases in which first phase is compulsory followed by the second phase in a such way a that the server may choose second phase with probability p or may return back to the system with probability 1−p and the vacation time follows general (arbitrary) distribution. The system may breakdown at random with mean break down rate α and repair process starts immediately in which the repair time follows exponential distribution with mean repair rate β. We obtain the time dependent probability generating functions in terms of th...
Non Markovian Queue with Two Types service Optional Re-service and General Vacation Distribution
2016
We consider a single server batch arrival queueing system, where the server provides two types of heterogeneous service. A customer has the option of choosing either type 1 service with probability p1 or type 2 service with probability p2 with the service times follow general distribution. After the completion of either type 1 or type 2 service a customer has the option to repeat or not to repeat the type 1 or type 2 service. As soon as the customer service is completed, the server will take a vacation with probability θ or may continue staying in the system with probability 1− θ. The re-service periods and vacation periods are assumed to be general. Using supplementary variable technique, the Laplace transforms of time dependent probabilities of system state are derived and thus we deduce the steady state results. We obtain the average queue size and average waiting time. Some system performance measures and numerical illustrations are discussed.
Single Server Bulk Queue with Second Optional Service, Balking and Compulsory Vacation
This paper is about the customers who arrive in bulk or group in a single server queueing system. The process takes place in Poisson distribution. The customers are provided with general services of two types in bulk of size M. The service is done on first come first serve basis in general distribution. Here the first service is essential and the second service is optional. Once the bulk gets serviced, the customer goes out of the system. The server compulsory goes to vacation when after completing the service. The vacation times are exponentially distributed. The server again goes to vacation or remain in the system if arrived bulk of customers is not enough to get the essential service. The server returns when arrived bulk is sufficient. The arriving batch balks during the period when the server is busy or when the server is on vacation or other constraints. This may result in the impatient behavior of the customers. We obtain the time dependent probability generating functions and from it the corresponding steady state results are derived. Also the average queue size and the system size and certain cases are explained.
A Heterogeneous Bulk Service Queueing Model With Vacation
2020
This paper is concerned with the study of bulk service M/M(a,b)/(2,1) queueing system of two heterogeneous servers with different service rates. In this model it is assumed that the arrival pattern is Poisson style with parameter λ and the service times are assumed to be mutually independent and exponentially distributed with parameters μ1and μ2 for the fast and slow servers respectively. The arrivals are served in batches according to FCFS discipline. In this model, the fast server (server 1) is always retained in the system and a delayed and single vacation policy for slow server (server 2)is discussed. The steady state solutions and the system characteristics are derived and analyzed for this model. The analytical results are numerically exemplified for different values of the parameters and levels also.
A Vacation Queue with Additional Optional Service in Batches
Applied Mathematical Sciences, 2009
A single server infinite capacity queueing system with Poisson arrival and exponential service time distribution along with second optional service in batches is considered. The server takes single vacation each time the system becomes empty and the duration of the vacation follows an exponential distribution. In steady state, the probability generating function for queue length has been obtained. The average queue length have been found and numerical results are presented to test the feasibility of the queueing model.
M[x]/G/1 Queue with Two Phase of Service and Optional Server Vacation
International Journal of Computer Applications, 2013
In this paper we analyze a single server queue with batch arrival Poisson input, two heterogeneous service with different general (arbitrary) service time distributions and two phase (compulsory and optional) server vacations with general (arbitrary) vacation period. The first phase of service is essential for all customers, as soon as the first service of a customer is completed, then with probability θ, he may opt for the second service or else with probability (1θ), he leaves the system. After completion of each service, the server will take compulsory vacation. The vacation period of the server has two heterogeneous phases. However, after returning from first compulsory vacation the server may take one more optional vacation with probability p or return back to the system with probability (1p). No server can take more than two vacations at a time. The probability generating function for the number of customers in the queue is found using the supplementary variable technique. The...
An M X / G /1 queueing system with disasters and repairs under a multiple adapted vacation policy
Naval Research Logistics (NRL), 2015
We consider a queueing system with batch Poisson arrivals subject to disasters which occur only when the server is busy and clear the system. Following a disaster the server initiates a repair period during which arriving customers accumulate without receiving service. The server operates under a Multiple Adapted Vacation policy. We analyze this system using the supplementary variables technique and obtain the probability generating function of the number of customers in the system in stationarity the fraction of customers who complete service, and the Laplace transform of the system time of a typical customer in stationarity. Finally, we examine a variation of the model in which the system is subject to disasters even when the server is taking a vacation or is under repair.